# How do I evaluate sin(pi/3) without using a calculator?

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69
Sep 25, 2016

$\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

#### Explanation:

Consider the series of diagrams below.

The interior angles of a triangle always add up to $\pi$ radians

An equilateral triangle will have all three angles $= \frac{\pi}{3}$ and all three sides of equal length.

If we set the sides of the equilateral triangle to $2$ we can avoid fractions when we divide the triangle into 2 halves vertically,

By the Pythagorean Theorem, the height (the side opposite the angle of $\frac{\pi}{3}$) will have a length of $\sqrt{3}$

$\sin \left(\frac{\pi}{3}\right) = \left(\text{side opposite "pi/3)/("hypotenuse}\right) = \frac{\sqrt{3}}{2}$

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